YES(O(1), O(n^2)) 9.17/2.84 YES(O(1), O(n^2)) 9.17/2.89 9.17/2.89 9.17/2.89 9.17/2.89 9.17/2.89 9.17/2.89 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 9.17/2.89 9.17/2.89 9.17/2.89
9.17/2.89
9.17/2.89
9.17/2.89
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
9.17/2.89 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
9.17/2.89 
↳2 CdtProblem
9.17/2.89 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
9.17/2.89 
↳4 CdtProblem
9.17/2.89 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
9.17/2.89 
↳6 CdtProblem
9.17/2.89 
↳7 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
9.17/2.89 
↳8 CdtProblem
9.17/2.89 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
9.17/2.89 
↳10 CdtProblem
9.17/2.89 
↳11 CdtKnowledgeProof (⇔)
9.17/2.89 
↳12 BOUNDS(O(1), O(1))
9.17/2.89
9.17/2.89 9.17/2.89
9.17/2.89
9.17/2.89

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

U11(tt, M, N) → U12(tt, activate(M), activate(N)) 9.17/2.89
U12(tt, M, N) → s(plus(activate(N), activate(M))) 9.17/2.89
U21(tt, M, N) → U22(tt, activate(M), activate(N)) 9.17/2.89
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N)) 9.17/2.89
plus(N, 0) → N 9.17/2.89
plus(N, s(M)) → U11(tt, M, N) 9.17/2.89
x(N, 0) → 0 9.17/2.89
x(N, s(M)) → U21(tt, M, N) 9.17/2.89
activate(X) → X

Rewrite Strategy: INNERMOST
9.17/2.89
9.17/2.89

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
9.17/2.89
9.17/2.89

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 9.17/2.89
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 9.17/2.89
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1)) 9.17/2.89
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 9.17/2.89
plus(z0, 0) → z0 9.17/2.89
plus(z0, s(z1)) → U11(tt, z1, z0) 9.17/2.89
x(z0, 0) → 0 9.17/2.89
x(z0, s(z1)) → U21(tt, z1, z0) 9.17/2.89
activate(z0) → z0
Tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 9.17/2.89
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 9.17/2.89
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 9.17/2.89
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 9.17/2.89
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.89
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
S tuples:

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 9.17/2.89
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 9.17/2.89
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 9.17/2.89
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 9.17/2.89
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.89
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

U11', U12', U21', U22', PLUS, X

Compound Symbols:

c, c1, c2, c3, c5, c7

9.17/2.89
9.17/2.89

(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 9 trailing tuple parts
9.17/2.89
9.17/2.89

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 9.17/2.89
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 9.17/2.89
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1)) 9.17/2.89
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 9.17/2.89
plus(z0, 0) → z0 9.17/2.89
plus(z0, s(z1)) → U11(tt, z1, z0) 9.17/2.89
x(z0, 0) → 0 9.17/2.89
x(z0, s(z1)) → U21(tt, z1, z0) 9.17/2.89
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.89
X(z0, s(z1)) → c7(U21'(tt, z1, z0)) 9.17/2.89
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 9.17/2.89
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 9.17/2.89
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1))) 9.17/2.89
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
S tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.89
X(z0, s(z1)) → c7(U21'(tt, z1, z0)) 9.17/2.89
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 9.17/2.89
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 9.17/2.89
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1))) 9.17/2.89
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

PLUS, X, U11', U12', U21', U22'

Compound Symbols:

c5, c7, c, c1, c2, c3

9.17/2.89
9.17/2.89

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

X(z0, s(z1)) → c7(U21'(tt, z1, z0)) 9.17/2.89
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
We considered the (Usable) Rules:

activate(z0) → z0 9.17/2.89
x(z0, 0) → 0 9.17/2.89
x(z0, s(z1)) → U21(tt, z1, z0) 9.17/2.89
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1)) 9.17/2.89
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 9.17/2.89
plus(z0, 0) → z0 9.17/2.89
plus(z0, s(z1)) → U11(tt, z1, z0) 9.17/2.89
U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 9.17/2.89
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
And the Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.89
X(z0, s(z1)) → c7(U21'(tt, z1, z0)) 9.17/2.89
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 9.17/2.89
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 9.17/2.89
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1))) 9.17/2.89
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation : 9.17/2.89

POL(0) = 0    9.17/2.89
POL(PLUS(x1, x2)) = 0    9.17/2.89
POL(U11(x1, x2, x3)) = [3] + [3]x1 + [3]x2 + [3]x3    9.17/2.89
POL(U11'(x1, x2, x3)) = x1    9.17/2.89
POL(U12(x1, x2, x3)) = [3] + x1    9.17/2.89
POL(U12'(x1, x2, x3)) = x1    9.17/2.89
POL(U21(x1, x2, x3)) = [3] + [3]x1 + [3]x2 + [3]x3    9.17/2.89
POL(U21'(x1, x2, x3)) = [1] + x1 + [4]x2    9.17/2.89
POL(U22(x1, x2, x3)) = [3] + [3]x1    9.17/2.89
POL(U22'(x1, x2, x3)) = x1 + [4]x2    9.17/2.89
POL(X(x1, x2)) = [4]x2    9.17/2.89
POL(activate(x1)) = x1    9.17/2.89
POL(c(x1)) = x1    9.17/2.89
POL(c1(x1)) = x1    9.17/2.89
POL(c2(x1)) = x1    9.17/2.89
POL(c3(x1, x2)) = x1 + x2    9.17/2.89
POL(c5(x1)) = x1    9.17/2.89
POL(c7(x1)) = x1    9.17/2.89
POL(plus(x1, x2)) = [3]    9.17/2.89
POL(s(x1)) = [2] + x1    9.17/2.89
POL(tt) = 0    9.17/2.89
POL(x(x1, x2)) = 0   
9.17/2.89
9.17/2.89

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 9.17/2.89
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 9.17/2.89
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1)) 9.17/2.89
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 9.17/2.89
plus(z0, 0) → z0 9.17/2.89
plus(z0, s(z1)) → U11(tt, z1, z0) 9.17/2.89
x(z0, 0) → 0 9.17/2.89
x(z0, s(z1)) → U21(tt, z1, z0) 9.17/2.89
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.89
X(z0, s(z1)) → c7(U21'(tt, z1, z0)) 9.17/2.89
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 9.17/2.89
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 9.17/2.89
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1))) 9.17/2.89
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
S tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.89
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 9.17/2.89
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 9.17/2.89
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
K tuples:

X(z0, s(z1)) → c7(U21'(tt, z1, z0)) 9.17/2.89
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1)))
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

PLUS, X, U11', U12', U21', U22'

Compound Symbols:

c5, c7, c, c1, c2, c3

9.17/2.89
9.17/2.89

(7) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0))) 9.17/2.89
X(z0, s(z1)) → c7(U21'(tt, z1, z0))
9.17/2.89
9.17/2.89

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 9.17/2.89
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 9.17/2.89
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1)) 9.17/2.89
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 9.17/2.89
plus(z0, 0) → z0 9.17/2.89
plus(z0, s(z1)) → U11(tt, z1, z0) 9.17/2.89
x(z0, 0) → 0 9.17/2.89
x(z0, s(z1)) → U21(tt, z1, z0) 9.17/2.89
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.89
X(z0, s(z1)) → c7(U21'(tt, z1, z0)) 9.17/2.89
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 9.17/2.89
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 9.17/2.89
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1))) 9.17/2.89
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
S tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.89
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 9.17/2.89
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
K tuples:

X(z0, s(z1)) → c7(U21'(tt, z1, z0)) 9.17/2.89
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1))) 9.17/2.89
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

PLUS, X, U11', U12', U21', U22'

Compound Symbols:

c5, c7, c, c1, c2, c3

9.17/2.89
9.17/2.89

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
We considered the (Usable) Rules:

activate(z0) → z0 9.17/2.89
x(z0, 0) → 0 9.17/2.89
x(z0, s(z1)) → U21(tt, z1, z0) 9.17/2.89
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1)) 9.17/2.89
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 9.17/2.89
plus(z0, 0) → z0 9.17/2.89
plus(z0, s(z1)) → U11(tt, z1, z0) 9.17/2.89
U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 9.17/2.89
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
And the Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.90
X(z0, s(z1)) → c7(U21'(tt, z1, z0)) 9.17/2.90
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 9.17/2.90
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 9.17/2.90
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1))) 9.17/2.90
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation : 9.17/2.90

POL(0) = [3]    9.17/2.90
POL(PLUS(x1, x2)) = x2    9.17/2.90
POL(U11(x1, x2, x3)) = [3] + [3]x1 + [3]x2 + [3]x3 + [3]x32 + [3]x2·x3 + [3]x1·x3 + [3]x12 + [3]x1·x2 + [3]x22    9.17/2.90
POL(U11'(x1, x2, x3)) = [2] + [3]x1 + x2 + [3]x1·x3 + [3]x12 + [3]x1·x2    9.17/2.90
POL(U12(x1, x2, x3)) = [3] + [3]x1 + [3]x12    9.17/2.90
POL(U12'(x1, x2, x3)) = [3]x1 + x2 + [3]x12    9.17/2.90
POL(U21(x1, x2, x3)) = [3] + [3]x1 + [3]x2 + [3]x3 + [3]x32 + [3]x2·x3 + [3]x1·x3 + [3]x12 + [3]x1·x2 + [3]x22    9.17/2.90
POL(U21'(x1, x2, x3)) = [3]x1 + x3 + x2·x3 + [3]x1·x3 + [3]x12 + [3]x1·x2    9.17/2.90
POL(U22(x1, x2, x3)) = [3] + [3]x1 + [3]x12    9.17/2.90
POL(U22'(x1, x2, x3)) = [3]x1 + x3 + x2·x3 + [3]x1·x3 + [3]x12    9.17/2.90
POL(X(x1, x2)) = x1·x2    9.17/2.90
POL(activate(x1)) = x1    9.17/2.90
POL(c(x1)) = x1    9.17/2.90
POL(c1(x1)) = x1    9.17/2.90
POL(c2(x1)) = x1    9.17/2.90
POL(c3(x1, x2)) = x1 + x2    9.17/2.90
POL(c5(x1)) = x1    9.17/2.90
POL(c7(x1)) = x1    9.17/2.90
POL(plus(x1, x2)) = [3]    9.17/2.90
POL(s(x1)) = [2] + x1    9.17/2.90
POL(tt) = 0    9.17/2.90
POL(x(x1, x2)) = 0   
9.17/2.90
9.17/2.90

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1) → U12(tt, activate(z0), activate(z1)) 9.17/2.90
U12(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 9.17/2.90
U21(tt, z0, z1) → U22(tt, activate(z0), activate(z1)) 9.17/2.90
U22(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 9.17/2.90
plus(z0, 0) → z0 9.17/2.90
plus(z0, s(z1)) → U11(tt, z1, z0) 9.17/2.90
x(z0, 0) → 0 9.17/2.90
x(z0, s(z1)) → U21(tt, z1, z0) 9.17/2.90
activate(z0) → z0
Tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.90
X(z0, s(z1)) → c7(U21'(tt, z1, z0)) 9.17/2.90
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1))) 9.17/2.90
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 9.17/2.90
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1))) 9.17/2.90
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)))
S tuples:

PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.90
U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
K tuples:

X(z0, s(z1)) → c7(U21'(tt, z1, z0)) 9.17/2.90
U21'(tt, z0, z1) → c2(U22'(tt, activate(z0), activate(z1))) 9.17/2.90
U22'(tt, z0, z1) → c3(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0))) 9.17/2.90
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
Defined Rule Symbols:

U11, U12, U21, U22, plus, x, activate

Defined Pair Symbols:

PLUS, X, U11', U12', U21', U22'

Compound Symbols:

c5, c7, c, c1, c2, c3

9.17/2.90
9.17/2.90

(11) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

U12'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0))) 9.17/2.90
PLUS(z0, s(z1)) → c5(U11'(tt, z1, z0)) 9.17/2.90
U11'(tt, z0, z1) → c(U12'(tt, activate(z0), activate(z1)))
Now S is empty
9.17/2.90
9.17/2.90

(12) BOUNDS(O(1), O(1))

9.17/2.90
9.17/2.90
9.87/2.94 EOF